I'm working with the Clifford algebra $ \text{CL}(1,1) $ and attempting to find an appropriate $ 2 \times 2 $ matrix representation. This algebra corresponds to a space with the metric signature $ (1, -1) $ or $ (-1, 1) $, and its generators should satisfy specific relations:
- $ (\gamma^0)^2 = I $ (the identity matrix)
- $ (\gamma^1)^2 = -I $ (the negative of the identity matrix)
- $ \{\gamma^0, \gamma^1\} = 0 $ (they anticommute)
I've explored some representations, like:
- $ \gamma^0 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} $
- $ \gamma^1 = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} $
However, in this representation, the pseudoscalar $ i = \gamma^0\gamma^1 $ doesn't square to -1 as expected. Could someone provide the correct $ 2 \times 2 $ matrix representation for $ \text{CL}(1,1) $ that adheres to these algebraic requirements?